Strongly Commuting Ring and The Prounet-Tarry-Escott Problem

In 1935, Wright conjectured that ideal solutions to the PTE problem in Diophantine number theory should exist. In this paper, we prove Wright's conjecture holds true based on the the representation theory of the minuscule strongly commuting ring introduced by Kostant in 1975 and the complex coefficient cohomology ring structures of the Grassmannian variey

The Prouhet-Tarry-Escott problem

Given natural numbers n and k, with n>k, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of Z, say X={x_1,...,x_n} and Y={y_1,...,y_n}, such that x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for i=1,...,k. Many partial solutions to this problem were found in the late 19th century and early 20th century. When k=n-1, we call a solution X=(n-1)Y ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. This thesis focuses on the ideal case. We extend the framework of the problem to number fields, and prove generalizations of results from the literature. This information is used along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers. We also extend a computation from the literature and find new lower bounds for the constant C_n associated to ideal PTE solutions. Further, we present a new algorithm that determines whether an ideal PTE solution with a particular constant exists. This algorithm improves the upper bounds for C_n and in fact, completely determines the value of C_6. We also examine the connection between elliptic curves and ideal PTE solutions. We use quadratic twists of curves that appear in the literature to find ideal PTE solutions over number fields

On the Reconstruction of Static and Dynamic Discrete Structures

We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the `classical' task of reconstructing finite point sets in $\mathbb{R}^d$). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems

Application of The Circle Method in Five Number Theory Problems

This thesis consists of three applications of the circle method in number theory problems. In the first chapter, we study a question of Graham. Are there infinitely many integers $n$ for which the central binomial coefficient $\binom{2n}{n}$ is relatively prime to $105 = 3\cdot 5 \cdot 7$? By Kummer's Theorem, this is the same to ask if there are infinitely many integers $n$, so that $n$ added to itself base $3$, $5$, or $7$, has no carries. A probabilistic heuristic of Pommerance predicts that there should be infinitely many such integers $n$. We establish a result of statistical nature supporting Pommerance's heuristic. The proof consists of the Fourier analysis method, as well as geometrically bypassing an old conjecture about the primes. In the second chapter, we discover an unexpected cancellation on the sums involving the exponential functions. Applying this theorem on the first terms of the Ramanujan-Hardy-Rademacher expansion for the partition function gives us a natural proof of a ``weak" pentagonal number theorem. We find several similar upper bounds for many different partition functions. Additionally, we prove another set of ``weak" pentagonal number theorems for the primes, which allows us to count the number of primes in certain intervals with small error. Finally, we show an approximate solution to the Prouhet-Tarry-Escott problem using a similar technique. The core of the proofs is an involved circle method argument. The third chapter of this thesis is about endpoint scale independent $\ell^p-$improving inequality for averages over the prime numbers. The primes are almost full-dimensional, hence one expects improving estimates for all $p >1$. Those are known, and relatively easy to establish. The endpoint estimates are far more involved however, engaging for instance Siegel zeros, in the unconditional case, and the Generalized Riemann Hypothesis (GRH) in the general case. Assuming GRH, we prove the sharpest possible bound up to a constant. Unconditionally, we prove the same inequality up to a logarithmic factor. The proof is based on a circle method argument, and utilizing smooth numbers to gain additional control of Ramanujan sums.Ph.D

An Algebraic Framework for Discrete Tomography: Revealing the Structure of Dependencies

Discrete tomography is concerned with the reconstruction of images that are defined on a discrete set of lattice points from their projections in several directions. The range of values that can be assigned to each lattice point is typically a small discrete set. In this paper we present a framework for studying these problems from an algebraic perspective, based on Ring Theory and Commutative Algebra. A principal advantage of this abstract setting is that a vast body of existing theory becomes accessible for solving Discrete Tomography problems. We provide proofs of several new results on the structure of dependencies between projections, including a discrete analogon of the well-known Helgason-Ludwig consistency conditions from continuous tomography.Comment: 20 pages, 1 figure, updated to reflect reader inpu